1. Introduction

Topography is of high importance in triggering and affecting meteorological processes in the atmospheric boundary layer. Orographic precipitation is probably the most evident yet poorly understood phenomenon in which topographic factors play the major role (Roe, 2005). Orographic enhancement of precipitation is a complex phenomenon, met at various spatial and temporal scales, known to trigger convective precipitation but also to determine long term precipitation patterns. Its characterization at various spatial scales is important for studying the involved processes (Gray and Seed, 2000), for downscaling of numerical weather prediction models (NWP, Baldauf and Schulz (2004)), for improving the orographic precipitation nowcasting schemes (Panziera et al., 2011) and for building real-time hydrological decision support systems (Germann et al., 2009).

Orographic precipitation is often related to the presence of a mountain range perpendicular to the prevailing flow direction which forces air to lift and to condense. In such situations higher precipitation rates are met on the windward flank and a shadowing effect is found on the leeward flank with respect to the main ridge (Figure 1, top), but, orographic effects are also present at much smaller spatial scales, enhancing and triggering precipitation. Triggering of thunderstorms on upwind slopes (Figure 1, centre) and at the top of hills by thermally induced flows (Figure 1, bottom) are two common mechanisms generating orographic precipitation at smaller spatial scales.

1.1. Numerical studies of orographic precipitation

The number and complexity of the contributing processes involved (see some examples in Roe, 2005) have been the subject of many numerical studies (Rotunno and Ferretti, 2001; Kunz and Kottmeier, 2006).

Physical-numerical models of orographic precipitation can be basically divided into diagnostic and prognostic types. The former are computed under given steady large-scale atmospheric conditions (flow direction, instability). However, they are less suited for modelling transient processes. Their calibration on limited datasets is hardly adjustable to different mountain ranges, different seasons and storm conditions (Barros and Lettenmaier, 1994). Such limitations are partly resolved by using prognostic models which allow integrating the full set of equations governing atmospheric processes. Prognostic models are particularly targeted to perform extensive numerical experiments but suffer from the computational burden.

A way out of these limitations is the analysis of extensive datasets provided by the operational use of mesoscale models. While physical mechanisms which favour the formation of orographic precipitation are known (Figure 1), orographic effects provoke strong systematic errors both on the windward and the lee sides of terrain features. This bias is found in many mesoscale NWP models (Baldauf and Schulz, 2004). While ongoing large-scale research programmes (Wulfmeyer et al., 2008) are expected to provide observational support for modelling multi-scale orographic effects, statistical methods may provide a useful solution to downscale the NWPs to sub-grid distances and to give an insight to a better understanding of the phenomenon in general.

The multi-scale nature of orographic precipitation and its inherent complexity implies that no single topographic index can provide sufficient correlation with observed precipitation even for modelling monthly averages (Faulkner and Prudhomme, 1998). A common approach is to bring together a set of topographic indices/features from the digital elevation model (DEM) based on physics-related heuristics. Such indices may include maximum drop within some relevant geographical vicinity, average slope, roughness index and bumpiness (Prudhomme and Reed, 1999) and other characteristics indicating if the terrain is favourable for orographic lift of the air masses.

Regression models often identify statistically significant influence of topographic indices, but the percent of variance explaining precipitation patterns is relatively low (Prudhomme and Reed, 1999). Some tools for improving spatial predictions of meteorological variables which face strong relationship with topography have been developed in machine learning, including the methods that facilitate the identification of the main contributing variables (Pozdnoukhov et al., 2009; Foresti et al., 2011b). The approaches that describe topographic conditions relevant to the orographic enhancement of precipitation still need to be developed.

1.3. Contributions of this work

The objective of this paper is to establish a methodology to study the orographic enhancement of precipitation using radar images and digital elevation models. The computation of terrain features such as gradients and convexities from a DEM provides a rich set of explanatory variables for orographic enhancement. The derivatives with respect to particular flow directions are further derived by combining the air mass motion vector field with terrain gradient. The computation of terrain attributes at specific spatial scales opens a way to reveal the one at which the phenomenon of orographic enhancement is mainly encountered. A key step in the proposed methodology is to detect the relevant precipitation cells from radar imagery and match them with the local topographic shapes responsible for enhancement processes. The resulting dataset is first analysed using a recently developed clustering technique to discriminate and select large scale flow conditions prone to orographic enhancement. The analysis of the repeatability of precipitation patterns at smaller spatial scales relative to the shape of topography provides a support for building computational data-driven models of susceptibility of orographic enhancement of precipitation.

Section '2. Methodology' of the paper outlines the methodology, Section '3. Data preparation' explains extensively the extraction of topographic descriptors from the DEM, the estimation of the motion vector fields and the detection of precipitation cells from radar imagery. In Section '4. Exploratory analysis of precipitation cells' typical flow conditions characterizing precipitation events and patterns of precipitation repeatability are explored using spectral clustering (SC). Section '5. Conclusion' discusses the advantages and limitations of the proposed methodology and concludes the paper.

2. Methodology

The proposed methodology for studying orographic enhancement using radar images and DEM consists of the following steps:

compute a set of topographical descriptors xDEM from the DEM which may relate to orographic lift such as slopes and convexity indicators as suggested by physical heuristics (Figure 1); repeat the operation for different spatial scales (Section '3.1. Feature extraction from DEM');

3. Data preparation

This section presents in detail the preparation of the dataset by processing of the radar images along with the DEM.

3.1. Feature extraction from DEM

It is essential to characterize the topographic factors properly in order to study the related orographic precipitation patterns. The Swiss DEM with a resolution of 250 × 250 m2 used for feature extraction is illustrated in Figure 2. A variety of topographic indices can be computed from it using convolution filters (Wilson and Gallant, 2000). The set of features is computed on the 1 × 1 km2 grid of the radar.

Figure 2. Swiss DEM of a 250 × 250 m resolution. The three radars are marked with the antenna sign

The extraction of a set of baseline topographic descriptors at different characteristic length scales (degrees of smoothness) was selected. A resulting set of correlated variables characterizes the geometric properties of the surface in the vicinity of every particular spatial location. Two sets of features have been considered (Foresti et al., 2011b):

the first set of features is computed by evaluating terrain gradient at a number of different spatial scales. The set of terrain gradients is computed from the DEM by gradually increasing the bandwidths of the smoothing Gaussian kernels. Gradient vectors will be combined with flow direction to compute flow directional derivatives (FD, see Section '3.2. Motion vector field and flow derivatives'). They are expected to explain stable upslope ascent and convection due to mechanical lifting;

the second set of topographic features was created using a combination of Gaussian smoothing filters. By subtracting two smoothed DEM surfaces obtained with different smoothing bandwidths, the ridges and valleys of different characteristic length scales are highlighted. These features are referred below to as Differences of Gaussians, DoG. DoGs are finite difference approximations of the Laplacian operator (Marr and Hildreth, 1980) which enables the computation of terrain convexity at different spatial scales. These features are expected to explain orographically triggered thunderstorms at the top of mountains.

Two sample features computed at different scales from each group are shown in Figure 3 along with the bandwidths σ of the convolution filter. Table 1 depicts all spatial scales at which features are extracted. Dubbed 2σ values are listed since they better represent the geographical area over which the DEM values are averaged.

Bold values depict the length scales of features which will be used for constructing scatterplots in Figure 10a and Figure 10b.

DoGs—1

0.6

1.3

2.1

2.9

5.4

9.6

25.4

DoGs—2

0.9

1.6

5.4

8.8

16.3

24.6

75.4

FD

0.2

1.0

2.0

5.1

10.2

20.4

102.2

3.2. Motion vector field and flow derivatives

Orographic precipitation is a dynamic process mostly driven by the flow field. It cannot be completely described by the static features listed in Section '3.1. Feature extraction from DEM' The motion vector field (flow direction and velocity) derived from subsequent radar images is a way of considering the spatial dynamics of precipitation which is commonly used in Lagrangian extrapolation-based nowcasting schemes (Germann and Zawadzki, 2002; Bowler et al., 2004).

Most of the motion vector field estimation techniques, also referred to as optical flow algorithms, are limited by two constraints (Sun et al., 2008): brightness constancy and spatial smoothness. Brightness constancy constraint assumes that optical features (for example a precipitation cell) persist over time and that their intensity is invariant by translation. Spatial smoothness constraint forces neighbouring pixels to have similar motion characteristics since they are likely to be governed by similar physical forces. The first constraint is violated if precipitation cells are dissipated or created, for example, in convective situations. The second one is violated if two close precipitation cells are moving in completely different directions, which is rare. Despite these constraints, optical flow algorithms provide a basis for modelling the displacement of precipitation on radar images.

The model used in this study was recently developed and explained in Sun et al. (2008). It defines a probabilistic model of optical flow (motion vector field) as:

where R1 and R2 are two consecutive radar images, U is the motion vector field composed of the two components of the displacement vector (u,v), p(R2|U, R1;ΩD) is the data term responsible for brightness constancy, p(U|R1;ΩS) is the spatial term describing spatial smoothness and Ω acts as a regularization parameter that controls the smoothness trade-off. Several values of Ω were tested and the one which provided a good compromise between smoothness and precision was chosen. To increase the robustness of the approach, optical flow was computed on radar images which were thresholded at 10 mm h−1 and smoothed with a Gaussian filter (2σ = 2 km).

Optical flow was then combined with terrain gradient to compute directional derivatives with respect to flow direction (referred to as FD, flow derivatives). Such flow varying derivative is an extremely important attribute for highlighting the upwind flank of mountains (see Figure 4) since orographic precipitation has a strong directional dependence (Weston and Roy, 1994; Panziera and Germann, 2010). Flow derivative is computed as follows:

where ∇z(xgeo) is the gradient vector of the altitude field z computed at location xgeo = (X, Y) and u(xgeo, t) is the vector with cosine (west to east, u) and sine (south to north, v) components of the motion vector field computed at xgeo = (X, Y) at time t. Flow derivatives are derived at a number of different spatial scales. Samples are shown in Figure 5(a) and (b) (the corresponding flow field is depicted in Figure 5(c) and (d): vectors are only displayed at the location of precipitation cells).

Figure 4. Flow derivative is positive if the precipitation cell is located on the upwind side, around 0 if moving parallel to the mountain ridge or over flatlands and negative if descending on the downwind side

Figure 5. (a) Flow derivative computed at a medium scale (2σ = 2 km) and the largest scale (2σ = 102.2 km) for prevailing southwesterly flows. The large scale derivative is displayed in a panel in the upper left corner but has the same spatial extension of the small scale derivative. Values are standardized to zero mean and unit variance. (b) Same as (a) but with northeasterly flows. (c) Example of a dynamic precipitation pattern: thunderstorms are moving over Swiss plateau with southwesterly winds; in the south of Grisons (see Figure 2 for details) there are typical isolated showers triggered by thermal ascent over mountains. (d) Example of a blocking precipitation pattern from northeast: precipitation cells move towards the Prealps but the southern leeward side of Alps is sheltered and no precipitation is observed. (e) and (f) Corresponding precipitation anomalies computed with Differences of Gaussians from which cells are detected. This figure is available in colour online at wileyonlinelibrary.com/journal/met

In summary, the space of features is composed of 10 fixed input features (X, Y, Z and 7 DoG) and 10 varying input features (7 FD, u, v and the rainfall intensity as estimated by radar). These features will be used for solving different tasks. Clustering will be performed in the 3D space composed of (u,v) and the largest scale flow derivative which basically describes the cells' position relative to the main Alpine divide (windward and leeward). All features excepting (u, v) and X, Y coordinates will be used in a future study to characterize orographic enhancement using non-parametric classification models. X, Y coordinates are registered for visualization purposes but are not directly used as explanatory variables.

3.3. Why different spatial scales?

The reasons why topographic and flow related features are extracted at different scales are of different origins. The first is due to the dynamics of airflow which is delayed in space and in time in relation to topographic forcing. Topography is not directly affecting precipitation but provokes saturation by forcing the uplift of an air mass. Due to cloud micro-physics (see a thorough description in Houze (1993)) there is a spatial and temporal delay between air saturation and precipitation. Also, numerical studies have demonstrated that the trajectory of hydrometeors is a function of their size and composition (snow flakes, rain drops) which in turn determines the fall speed (Hobbs et al., 1973). Hydrometeor fall speed and wind intensity are just two among many controlling factors characterizing the spatial disagreement between the place of air uplift, condensation and the location of precipitation fallout at the ground plane.

Heavy precipitation is more likely to occur on the upwind flank at a certain spatial scale, and is the subject of linear models of orographic precipitation (Smith and Barstad, 2004). However, a too small-scale flank of a ridge perpendicular to flow direction is often not enough to trigger the mechanism of condensation and precipitation and hence is a poor explanatory variable. Larger scale features are expected to be more informative since the uplift of air needs a slope of a certain size and length to take place significantly.

Airflow speed and air instability are also controlling factors for the spatial distribution of precipitation. Airflow speed is proportional to the activity of a cold front approaching a topographic barrier which in turn increases precipitation quantities on upwind slopes (Johansson and Chen, 2003). Unstable air associated with strong turbulent flows increases the efficiency of particle growth and fallout causing higher precipitation rates on upwind slopes (Houze et al., 2001; Panziera and Germann, 2010).

The relation between wind speed and the spatial distribution of precipitation is even more complex. High precipitation rates can be found at the top of hills and ridges (described by DoG) as modelled by Smith and Barstad (2004) and not on upwind slopes depending on the scale of analysis and the wind speed. There are also cases when precipitation is enhanced on the leeside of a mountain due to lee-side convergence behind small topographic obstacles (Cosma et al., 2002), to spillover behind narrow mountains (Smith and Barstad, 2004), to gravity waves and lee-side cold air pools (Zängl, 2005), to the time elapsed between rain drop nucleation and falling to the ground (Roe, 2005; Zängl, 2007) which can be important when associated with seeder-feeder processes (Zängl, 2007). These effects are less frequent and need small scale explanatory features to be described. Extensive numerical simulations analysing the relative positioning of maximum rainfall rates with respect to a mesoscale ridge are illustrated in Miglietta and Rotunno (2009, 2010). The wind speed, air stability, height and width of the ridge determine the spatial distribution of high rainfall rates. Two real case studies, including the event of August 2005, are studied in Zängl (2007) using numerical simulations and observed rain gauge measurements. These show that windward and leeward accumulations of precipitation vary from region to region and depend on wind speeds, freezing levels and the presence of seeder-feeder mechanisms.

In the present paper, topographic characteristics describing the neighbourhood of a precipitation cell are considered by computing delocalized features which describe them at different spatial scales (Section '3.1. Feature extraction from DEM'). An alternative approach would be to integrate topographic data surrounding the location of interest for different distances and directions, but the number of input dimensions of features would increase dramatically.

Finally, Roe (2005) concludes that ‘precipitation maximizes over the windward slopes, whereas for smaller hills the maximum tends to occur nearer the crest’ and that there is ‘evidence of a close association between orography and precipitation patterns at spatial scales of a few kilometres’. The problem of analysing at which spatial scale these statements become valid is crucial for a better understanding of orographically induced precipitation.

The range of scales considered in this paper (see Table 1) is aimed to capture most of precipitation variability in the Alps due to topographic forcing. The minimum and maximum ranges of smoothing bandwidths are chosen to be wide enough to have an extensive feature set from which the statistically relevant ones can be selected. Thus, small scale features are expected to be explanatory variables for enhancement effects due to triggered convection. On the other hand, large scale features can explain widespread precipitation patterns due to blocking by the Alps.

3.4. Precipitation cells detection

Radar images illustrating the proposed methodology of orographic precipitation pattern analysis concern the Swiss Alps region in the period of 18–23 August 2005. The northern flank of the Alpine chain was affected by severe flooding due to persistent thunderstorms and blocking situations (Rotach et al., 2006).

Meteoswiss operates a network of three C-band Doppler weather radars located at the top of Monte Lema, La Dôle and the Albis (see Figure 2). Radar-derived composite instantaneous rainfall rates at a 1 × 1 km2 grid resolution and with 5 min of temporal resolution are available. They are derived with the pre-processing steps described in Joss et al. (1998) and Germann et al. (2006), including the corrections for vertical profiles in sheltered regions, application of ground clutter elimination algorithms, bias correction with respect to rain gauges and corrections for the bright band effect. It is currently the most reliable radar product available from Meteoswiss. However, profile corrections (Germann and Joss, 2002) cannot solve all visibility problems of Swiss radars in regions which are shielded by mountainous relief or by obstacles close to the antenna. In such places the lowest elevation scans are too high in the atmosphere for a reliable estimate of rainfall rates at the ground. Only convective precipitation presenting a sufficient vertical extension can be detected, while stratiform precipitation is often unseen. The visibility of Swiss radars can be evaluated by geometric approaches to correct radar-rain gauge biases using the distance from the radar, the height of the lowest visible scan and the height of the ground in linear or non-linear regression models (Gabella and Perona, 1998; Gabella et al., 2001, 2005; Golz et al., 2005). The effective radar visibility can then be observed by evaluating the relative rainfall detection rates of radar and rain gauges in the spatial domain (Wüest et al., 2010). In the Valais and Grisons regions (see Figure 1) the height of the lowest visible scans varies between 4000 and 6000 m above sea level which does not allow detecting and extracting precipitation patterns (see Golz et al., 2005, a detailed map of the height of the lowest scans of the Monte Lema radar). The analysis is restricted to the places monitored by low level scans such as the Prealps which were also the most touched by orographic precipitation during August 2005, but low level scans are more prone to ground clutter mainly due to the backscattering of beam by orographic features which eventually overestimates radar reflectivity. This has to be considered when extracting and analysing precipitation fields along with the interpretation of results.

The first step in the analysis of orographic enhancements is the extraction of precipitation cells from radar images. There are well-established algorithms for convective cell detection and tracking (Dixon and Wiener, 1993; Lakshmanan et al., 2009). The operational precipitation cell tracking approach adopted in Switzerland (Thunderstorms Radar Tracking—TRT) is described in Hering et al. (2004). It is specifically targeted to extract and track convective cells characterized by significant vertical and spatial extensions.

The approach proposed in the present paper considers the extraction of precipitation cells by finding points of extrema, above a certain threshold, of filtered radar images. Compared to TRT, it also allows extraction of non-convective cells, including the ones due to orographic enhancement effects. Rain rates of radar images were first filtered using DoGs with small and large scales (2σ = 2 km and 2σ = 20 km). The resulting image describes precipitation anomalies in the spatial domain such as isolated cells or precipitation enhancements inside spread precipitation areas. Points of extrema exceeding 5 mm h−1 of the field of precipitation anomalies have been extracted (Figure 5(e) and (f)). Filtering of the radar image is also performed to shrink the effects of clutter and ground echoes. While Germann et al. (2009) report that 98% of all cluttered pixels are eliminated, one should still be aware of the presence of noise and outliers provoked by beam and shadow effects of the radar. A rough check for ground clutter for the locations touched by many precipitation cells was carried out with respect to the accumulated precipitation field constructed by integrating all radar images. Only a few of the detected cells were located in places showing anomalously high values (due to the summation of clutter effects) compared to the climatology of the event.

The effective spatial resolution of the measured precipitation field is decreasing as a distance to the radar. The scanned volume is re-sampled to provide precipitation data on a regular grid of 1 × 1 km2. However, the real resolution of the precipitation field, and in particular for places far from the radar, is better captured by working with filtered radar images. Notice that under the considered setting we only aim at identifying the location and topographic shape below precipitation cells without requiring the reliable quantitative precipitation estimates.

The full set consists of 28 758 cells detected in 1728 radar images captured over 6 days. The instances in this dataset, along with their topographic and optical features (dimensions) result both from orographic and atmospheric effects. Some outliers are present in the dataset due to instrumental errors.

4. Exploratory analysis of precipitation cells

Through the exploratory analysis we show that the detected cells are not randomly distributed in geographical and feature space of topographical descriptors. The observed clustered structure show interesting regularities demonstrating that orographic enhancement effects are more likely found at specific geographical places with defined topographic conditions.

4.1. Spectral clustering

The most well known traditional clustering method is the k-means algorithm (Steinhaus, 1956). Due to its simplicity and easy interpretability it is an accepted standard as a sub-routine in practical clustering problems, involving for example those in meteorological studies related to weather types classification. However, k-means is, however, unable to delineate non-convex shapes and is known to perform poorly if input variables are correlated. Spectral clustering (Hagen and Kahng, 1992; Shi and Malik, 2000; Ng et al., 2001) is a clustering method overcoming the drawbacks of k-means. It is well suited for delineating non-linear cluster shapes such as low-dimensional manifolds in high-dimensional spaces.

In spectral clustering (SC), one considers the data samples as the nodes of a weighted graph, where only the nodes in a close proximity are linked, and applies the methods of spectral graph theory to study its inner structure. In this study the approach of Ng et al., (2001) is followed where the graph is studied via the analysis of the eigenvectors of the normalized graph Laplacian. The eigen-analysis identifies strongly inter-linked parts of the graph while discriminating those graph components connected by a small number of links (a schematic example is given in Figure 6). In practice, the k-means algorithm is applied in the spectral space composed of the first eigenvectors where the clusters present typical spherical shapes (Figure 6). This is the computationally intensive part of the algorithm compared to a simpler algorithm such as k-means.

Figure 6. Illustration of spectral clustering. The structure of the data graph in input space can be revealed by k-means applied on the eigenvectors of graph Laplacian, see Ng et al., (2001) for details

The use of spectral clustering is justified by the fact that optical flow descriptors computed at the geographical locations of the cells which move over mountainous terrain may form complex non-linear shapes due to the variations of flow conditions rather than spherical-shaped clusters (a typical example is shown in Figure 7).

4.2. Characterization of flow conditions using SC

During the 6 days of precipitation several flow conditions and atmospheric processes were observed in the Alpine region (Rotach et al., 2006). It is essential to distinguish the different meteorological conditions and especially to remove the ones which are less concerned by orographic effects. At the same time, the spatial distribution of orographic precipitation events is highly dependent on mesoscale flow conditions (Weston and Roy, 1994; Rotunno and Houze, 2007; Panziera and Germann, 2010) and motivates a preliminary stratification of the data depending on wind direction and intensity.

A common approach is to manually select weather situations presenting orographic effects on the windward side of mountains and then to stratify the data according to flow direction (Panziera and Germann, 2010). In the present case, spectral clustering is applied to find natural partitions in the set of cells with respect to (u,v) flow components and by the flow directional derivative computed with the largest scale (see an example in the top-left corners of Figure 5(a) and (b)). Such derivative reports if the cell originated on the windward (+ sign in Figure 5(a) and (b)) or on the leeward side (− sign in Figure 5(a) and (b)) of the Alps with respect to the flow direction characterizing its formation and motion.

Figure 7(a) shows cells in (u,v) wind components and flow derivative. Two important clusters of orographic cells are spotted in the (u,v) dimensions: one is characterized by wind spanning SW to SE directions (we will refer to southerly cluster) and the other is located from NW to NE wind directions (northerly cluster). A visual multi-scale analysis of the two given clusters reveals the presence of smaller clusters if the large-scale flow derivative is integrated as a third dimension. The southerly cluster may be divided again in four to five small regions and the northerly cluster in two large regions. A sequential hierarchical top-down application of spectral clustering is performed: the first one to detect the two big clusters and the following two are applied independently on each cluster to find intra-clusters. Spectral clustering was applied using five nearest neighbours for constructing the graph adjacency matrix. The total number of clusters has been chosen by inspecting the patterns of precipitation cells.

Figure 7(b) shows the detected clusters. The corresponding meteorological details for the clusters are summarized in Table 2. Figure 8 presents clusters in the original space of geographical X,Y coordinates.

Figure 8. Spatial distribution of the eight clusters over Switzerland. Details are given in Table 2

Potential conditions for orographic precipitation are met on windward flanks (positive flow derivatives, clusters NWw, NEw, SEw, SWw1, SWw2). Situations SWw1 and SWw2 are respectively the ones with strong and weak winds (see Table 2). On the contrary, cells developing on leeward side or parallel to the Alps are assumed being non-orographic events (clusters Slee, Ep and SWp).

The clusters detected in this paper can be easily compared with the weather types corresponding to similar mesoscale flows. For example, Fliri (1984) provides an extensive climatology of the distribution of precipitation in the Alps from rain gauge networks for different flow conditions and seasons. The present study found a correspondence between the spatial distribution of precipitation patterns in Fliri (1984) and the geographical location of cells within the clusters. However, the use of high resolution radar data gives additional spatial details unveiling preferred regions for cells development and occurrence which is not achievable using the coarse resolution of common rain gauge networks.

4.3. Repeatability of precipitation patterns

Triggering or enhancement of precipitation can occur both for convective and advective situations. Convective storms are known to be triggered by thermal winds, and particularly on the top of mountains (described by DoGs). Thunderstorms moving over topography can be enhanced if they reach a sufficiently large slope (described by the flow derivative). The advection of sufficiently moist air can lead to saturation causing precipitation by simple topographic forcing. On the other hand, if precipitation has already started before reaching a given location, it can be enhanced if orographic conditions are favourable. However, a major part of precipitation cells are not triggered or enhanced by topography but their behaviour is driven by atmospheric processes. For example, once a thunderstorm is triggered over a hill top the storm potentially becomes independent and touches various topographies in its path.

Therefore, a criterion defining which cells are influenced by topography is needed. The approach is based on the repeatability of precipitation patterns under similar flow conditions. Repeatability of precipitation patterns is measured by counting the number of times that a location faces a precipitation hot spot (cell) under given flow conditions (same cluster). Places presenting high values are more likely to be influenced by topography while other places are less concerned with topography. The counter on precipitation repeatability can be further summarized by applying an indicator function defining classes of orographic and non-orographic effects.

Figure 9 shows the spatial distribution of the indicator evaluated with a threshold of four counts. The very dynamic situation SWw1 was omitted from the analysis since very few repeated precipitation cells were found. Precipitation cells repeatability is more often found in the Prealps which was the region mostly affected by the precipitation event. Other repeatability effects can be spotted in Valais, south of Grisons, Valtellina and the Simplon region. The choice of the threshold defining the two classes is purely empirical. The trade-off between the number of cells which compose the orographic class and the evidence of class separability guided its selection. Too high a threshold leaves few cells in the orographic class whose cells are not representative of the statistical distribution and can be affected by measurement errors. Lower threshold values are desirable since they better represent the statistical distribution. Too low a threshold causes the mixing of the two classes which become unseparable unless robust pattern detection algorithms are used. Figure 10(a) and (b) plot the two classes on the space composed of topographic features and two distinct probability distributions can be depicted. While orographic cells are more likely to be found on negative convexities (top of hills) and on positive directional derivatives (upwind versants), the non-orographic cells are centred around the origin and represent more or less all kind of topographies. Non-orographic cells in Figure 10(b) are located on slightly positive flow derivatives since most of the cells were detected upstream of the Alpine ridge. The results confirm the outcomes of other orographic precipitation studies (Gray and Seed, 2000; Houze et al., 2001; Johansson and Chen, 2003; Anders et al., 2007).

Figure 9. Spatial distribution of orographic (black crosses) and non-orographic (white dots) classes. The well visible concentration of cells in the northern side of Grisons is due to the parallel orientation of the valley with respect to the direction of radar beam. It allows scanning the lower levels of the atmosphere where orographic enhancement is more accentuated. The convergence of flows to this valley provoking potential orographic precipitation is met in particular with northwest winds (Figure 8)

Figure 10. Orographic (black crosses) and non-orographic (grey dots) classes in (a) the space composed of features DoG (2σ1 = 5.4 km/2σ2 = 16.3 km) and small-scale flow derivative (2σ = 2 km) and (b) the space composed of features DoG (2σ1 = 5.4 km/2σ2 = 16.3 km) and large scale flow derivative (2σ = 102.2 km). The sudden decrease in the occurrence of orographic cells in the upper left side of (b) is due to the natural limit of the large scale derivative which maximizes in the middle of Prealps (the place with pronounced precipitation repeatability). The dashed line is a visual help for separating the two classes. Axes values are standardized to zero mean and unit variance. The visualized scales are marked in bold in Table 1

As already mentioned in Section '3.4. Precipitation cells detection', radar precipitation measurement in complex topography is principally limited by beam blocking and ground clutter. The former does not harm too much the interpretation of results because precipitation cells are simply not detected in shielded regions and are not saved in the datasets. The second is more problematic since it can bias the repeatability of precipitation towards higher counts. More research is needed to quantify the relative contribution of ground clutter and orographic precipitation in these places. On the other hand, the unmixing of these two components could be helped by the methodology proposed in this paper by detecting the locations showing too high repeatability counts which cannot be explained only by the shape of topography.

5. Conclusion

This section summarizes the paper, discusses the advantages of the methodology and lists potential developments.

5.1. Summary

Image processing and clustering techniques were applied to weather radar imagery and a digital elevation model to derive the conditioning topographic factors for orographic precipitation enhancement. In particular, terrain convexity and flow derivatives describing upwind slopes at multiple spatial scales were used to characterize the repeatability and persistence of precipitation cells extracted from the radar archive.

Spectral clustering was applied in a preliminary step to define clusters of similar flow conditions leading to the formation of potential orographic precipitation patterns. The joint use of the flow direction and the relative positioning of cells with respect to the main Alpine chain (windward or leeward) allowed discovering geographically structured clusters. This exploratory part aimed at finding similar weather types in terms of flow conditions and exposure to main Alpine flanks and facilitated the selection of events presenting potential orographic effects.

Within each cluster the cells were classified as orographic and non-orographic by using a threshold on precipitation repeatability. The geographical distribution of orographic precipitation cells was explained by the local terrain shape. By comparing the statistical distribution of orographic and non-orographic cells in the space of topographic features, patterns of cell repeatability were found mainly on upwind slopes and at the top of hills. On the other hand, non-orographic cells did not show a clear relation with the topography.

The results of this study demonstrate that the enhancement of rainfall due to orography, which leads to stationary and repetitive rainfall cells, occurs on scales of a few kilometres, at the edge between micro and meso-gamma scales.

5.2. Flexibility of the methodology

The choice of terrain attributes at a number of explicit spatial scales provides one of the main strengths of the methodology since it allows a thorough description of potential orographic effects for a given topographic feature. Wrong conclusions about the explanatory power of terrain features could be drawn if derived only at one specific spatial scale.

Clustering of cells allows discriminating an appropriate number of flow conditions for every particular dataset chosen for the analysis. It gives an interpretable summary for a precipitation event under study and allows a user expert to select particularly interesting precipitation and flow conditions for detailed analysis. The methodology is modular and allows for improved cells detection, clustering and motion vector field estimation methods to be applied at corresponding stages of the analysis, as well as for extended sets of explanatory topographic or meteorological features relevant to the geography of a region under study.

5.3. Future research directions

Future research directions consider the use of external variables characterizing atmosphere stability and further developments of image processing tools including the spatial extension and forecasting the evolution of the detected precipitation cells. Datasets for longer periods of observation have to be processed to represent a wider set of weather situations, flow directions and orographic precipitation patterns.

The modelling of orographic enhancement will be approached using robust techniques of classification and density estimation able to work with large and high-dimensional datasets (see preliminary results in Foresti et al., 2011a). The influence of the threshold defining orographic and non-orographic classes can be reduced by using robust classification methods allowing an explicit control over the model's complexity through the regularization of data misfits (cells falling on the other class). Their applicability is the subject of current studies.

A broader framework could be related to uncertainty estimation in hydrological context when generating ensembles of precipitation fields. One needs to estimate the covariance structure of the field (Germann et al., 2009) or involve pattern generation methods from multi-point statistics (Wójcik et al., 2009). In both approaches, orographic enhancement at different spatial scales is an effect still needed to be accounted for. Knowledge of orographic enhancement can be used to build advanced covariance models for ensemble generation or to condition sampling distributions on the topography within multi-point statistics framework.

Orographic enhancement maps characterizing how likely topographic and flow conditions produce repeatability effects on precipitation could also aid to evaluate the effectiveness of numerical models in complex terrain. Enhancement maps could be used to correct the outputs of numerical models by warping the relative distribution of precipitation between windward and leeward sides of terrain features at multiple scales to fit to the measured patterns of orographic precipitation.

Acknowledgements

The research is funded in part by the Swiss National Science Foundation projects GeoKernels: kernel-based methods for geo- and environmental sciences (Phase II) (No 200020-121835/1). A. Pozdnoukhov acknowledges the support of Science Foundation Ireland under the National Development Plan, particularly through Stokes Award and Strategic Research Cluster grant (07/SRC/I1168). We would like to thank Professor Mikhail Kanevski for the discussions on data modelling. We thank MeteoSwiss for providing radar data. We also acknowledge the Radar and Satellite team (RASA) at MeteoSwiss, in particular Dr Luca Panziera, Dr Marco Gabella, Dr Pradeep Mandapaka and Dr Urs Germann, for the data quality check (ground clutter), for the discussions about correction issues for radar visibility and for the interesting insights about the use of radar for nowcasting of orographic precipitation.

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